Ah - not the first time today I was tripped up by the rule that we only need to find a single set of xns with the maximum shattering. I keep forgetting and looking for some set of xns which doesn't.

So here's a method that will work for any N, and I don't even need all of your hypotheses; it's enough just to use the set of H's where n=2^k.
H0=+1 for 1,3,5,7,...
H1=+1 for 1,2,5,6,9,10,...
H2=+1 for 1,2,3,4,9,10,11,12,17,18,19,20,...
H3=+1 for 1,2,3,4,5,6,7,8,17,18,19,...

The easy way to see it is to use binary notation.
H0=+1 for any n with the last digit=1
H1=+1 for any n with the second-to-last digit=1
H2=+1 for any n with the third-to-last digit=1 etc.

Now gH(N)=2^N. Here's how we build our set of x1,...,xN that we can shatter.

For N, take the set Z of every combination of N 1s and 0s and use them to build our x1,...,xN. They will be very big numbers indeed, with 2^N digits or so.
The first digit of xi, call it xi1, is the ith digit of the first element of Z. If that is 000000...0 (N 0s), the first digits of all the xis will all be zero as well, so all of the numbers x1,...,xN will start with 0.
If the second element of Z is 000000..01 - might as well keep them in order - the second digit of x1,...,xN will be 0, 0, ..., 0, and 1 respectively. And so forth, for all 2^N digits.

These N very special binary numbers are shattered by H0 through H(2^N-1), as H(i) picks out the ith digit of each of our numbers, and the digits cover every possibility in Z.

This problem makes me wonder if the VC dimension formula for the growth function isn't (sometimes?) far too restrictive. It's true that the growth function here is 2^N, and therefore our problem is maybe un-learnable. But is that really true? Because we could find one wacko example with numbers 2^50 digits long, does that mean that our hypotheses are really much too general? Actually, this set of hypotheses is awfully restricted, and you (probably) can't hardly do anything with them for almost all sets of x1,...xn. Maybe we should have a modified version of the VC formula, where the bound works for almost all sets of xn. Is it still true that the growth will be polynomial, almost always?

This problem makes me wonder if the VC dimension formula for the growth function isn't (sometimes?) far too restrictive. It's true that the growth function here is 2^N, and therefore our problem is maybe un-learnable. But is that really true? Because we could find one wacko example with numbers 2^50 digits long, does that mean that our hypotheses are really much too general? Actually, this set of hypotheses is awfully restricted, and you (probably) can't hardly do anything with them for almost all sets of x1,...xn. Maybe we should have a modified version of the VC formula, where the bound works for almost all sets of xn. Is it still true that the growth will be polynomial, almost always?

Indeed, the VC dimension addresses a worst-case scenario. The advantage is that it is applicable regardless of what the input probability distribution turns out to be, so 'learnable' means guaranteed to be learnable. The disadvantage is that it may be unduly pessimistic in many practical cases. There is a modified version of this analysis based on expected values. It is more involved technically, and can be found in Vapnik's book.

__________________Where everyone thinks alike, no one thinks very much

Ah - I see that in your book (footnote p. 51), the case of convex regions is mentioned as an example where an "estimated" growth bound works. I'm guessing that's because even though points on the rim of a circle and such can be shattered, "almost every" set of N points is going to have some on the interior, which cannot be shattered.

In this example, Michael, it's not really correct to think of the shattered sets of points as being untypical. Almost all integers are very large! For example, given N if you pick a number M and choose a set of N points randomly in [M, 2M], the probability of the points not being shattered by this hypothesis set will tend to zero as M tends to infinity. [exercise for reader ]

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